Resolution Proofs as a Data Structure for Logic Synthesis

1. Resolution Proofs as a Data Structure for Logic Synthesis John Backes (back0145@umn.edu) Marc Riedel (mriedel@umn.edu) Electrical and Computer Engineering University of Minnesota

2. Data Structures • Sum of Products (SOPs) • Advantages: explicit, readily mapable. • Disadvantages: not scalable. • Binary Decision Diagrams (BDDs) • Advantages: canonical, easily manipulated. • Disadvantages: not readily mapable, not scalable.

3. Data Structures • And Inverter Graphs (AIGs) • Advantages: • Compact. • Easily convertible to CNFs. • Scalable, efficient. • Disadvantages: • Hard to perform large structural changes.

4. Resolution Proofs • Implicitly extracted from SAT solvers; converted to logic via Craig Interpolation. • Utilize as a data structure to perform logic manipulations. • Advantages: • Scalable, efficient. • Can effect large structural changes.

5. AIG Synthesis • Re-writing • Cuts are replaced by pre-computed optimal structures (Mishchenko ’06). • SAT Sweeping • Nodes of an AIG can be merged by proven equivalence (Zhu ‘06). • SAT-Based Resubstitution • Target nodes are recomputed from other nodes (Lee ‘07).

6. AIG Synthesis SAT-Sweeping (merging equivalent nodes)

7. AIG Synthesis SAT-Sweeping (merging equivalent nodes)

8. AIG re-writing Local manipulations performed on windows Local minimums can be reached AIG Synthesis

9. AIG re-writing Local manipulations performed on windows Local minimums can be reached AIG Synthesis

10. AIG re-writing Local manipulations performed on windows Local minimums can be reached AIG Synthesis

11. Resubstitution a.k.a. Functional Dependencies Giventarget: f (z1,z2,…,zn), Given candidates:x1(z1,z2,…,zn), x2(z1,z2,…,zn), …, xm(z1,z2,…,zn) is it possible to implement f (x1,x2,…,xm)?

12. Resubstitution f (x1,x2)? Aig Synthesis

13. Resubstitution f (x1,x2)? Large changes This question is formulated as a SAT instance. Craig Interpolation provides implementation. Aig Synthesis

14. Craig Interpolation • Given formulas Aand Bsuch that A→¬B, there exists I such that A→ I →¬B • I only contains variables that are present in both Aand B. I B A

15. Craig Interpolation SAT?: ( f )(CNFLeft)( f *)(CNFRight)(x1 = x1*)(x2 = x2*)…(xm = xm*) A B • f (x1,x2,…,xm)? • If UNSAT, a proof of unsatisfiablility is generated. • An implementation of f is generated from the proof. (Lee ‘07)

16. Resolution Proofs • A proof of unsatisfiability for an instance of SAT forms a graph structure. • The original clauses are called the roots and the empty clause is the only leaf. • Every node in the graph (besides the leaves) is formed via Boolean resolution. • E.g.,: (c + d)(¬c + e)→(d + e)

17. A Resolution Proof Clauses of Aare shown in red, and clauses of Bare shown in blue. ( )( )( )( )( ) ( )( ) a + ¬ c + d ¬ a + ¬ c + d a + c ¬ a + c ¬ d d + ¬c a + b ( ) ( ) c ¬ c ( )

18. Example: Generating I ( )( )( )( )( ) ( )( ) a + ¬ c + d ¬ a + ¬ c + d a + c ¬ a + c ¬ d d + ¬c a + b ( ) ( ) c ¬ c ( )

19. ( )( )( ) ( ) a + c ¬ a + c ¬ d d + ¬c ( ) ( ) c ¬ c ( ) Example: Generating I

20. a c ¬a c Example: Generating I ( )( )( ) ( ) a + c ¬ a + c ¬ d d + ¬c ( ) ( ) c ¬ c ( )

21. Example: Generating I a c ¬a c )( ) ( ) ¬ d d + ¬c ( ) ( ) c ¬ c ( )

22. ( ) ¬ d Example: Generating I a c ¬a c ( ) ( ) c ¬ c ( )

23. ( ) ¬ d Example: Generating I a c ¬a c ¬d ( )

24. Generating Multiple Dependencies • Often, goal is to synthesize dependencies for multiple functions with overlapping support sets. • In this case, multiple proofs are generated and then interpolated.

25. Large portions of a network can be converted to a resolution proof. fj(x1,x2,x3,x4,x5,x6)? fk(x1,x2,x3,x4,x5,x6)? Example

26. Example fj(x1,x2,x3,x4,x5,x6)? fk(x1,x2,x3,x4,x5,x6)?

27. Example fj(x1,x2,x3,x4,x5,x6)? fk(x1,x2,x3,x4,x5,x6)?

28. Observation • There are often many ways to prove a SAT instance unsatisfiable. • Same/similar nodes shared between different proofs.

29. Example fj(x1,x2,x3,x4,x5,x6)? fk(x1,x2,x3,x4,x5,x6)?

30. Example fj(x1,x2,x3,x4,x5,x6)? fk(x1,x2,x3,x4,x5,x6)?

31. Restructuring Mechanism • Some clause c can be resolved from some set of clauses Wiff(W)(c) is unsatisfiable. • The resolution proof of (W)(c)can be altered to show how ccan be resolved from W. (Gershman ‘08)

32. Example Can (a + b) be resolved from (a + e + d)(a + b + d) (a + b + d + e)? (Gershman ‘08)

33. Example Can (a + b) be resolved from (a + e + d)(a + b + d) (a + b + d + e)? (Gershman ‘08)

34. Example Can (a + b) be resolved from (a + e + d)(a + b + d) (a + b + d + e)? (Gershman ‘08)

35. Example Can (a + b) be resolved from (a + e + d)(a + b + d) (a + b + d + e)? (Gershman ‘08)

36. Example Can (a + b) be resolved from (a + e + d)(a + b + d) (a + b + d + e)? (Gershman ‘08)

37. Example Can (a + b) be resolved from (a + e + d)(a + b + d) (a + b + d + e)? (Gershman ‘08)

38. Example Can (a + b) be resolved from (a + e + d)(a + b + d) (a + b + d + e)? (Gershman ‘08)

39. Example Can (a + b) be resolved from (a + e + d)(a + b + d) (a + b + d + e)? (Gershman ‘08)

40. Example Can (a + b) be resolved from (a + e + d)(a + b + d) (a + b + d + e)? (Gershman ‘08)

41. Example Can (a + b) be resolved from (a + e + d)(a + b + d) (a + b + d + e)? (Gershman ‘08)

42. Example Can (a + b) be resolved from (a + e + d)(a + b + d) (a + b + d + e)? (Gershman ‘08)

43. Example Can (a + b) be resolved from (a + e + d)(a + b + d) (a + b + d + e)? (Gershman ‘08)

44. Proposed method • Select potential target functions with the same support set: f1(x1,x2,…,xm), f2(x1,x2,…,xm), … , fn(x1,x2,…,xm) • Generate collective resolution proof. • Structure the proofs so that there are moreshared nodes.

45. Which nodes can be shared? • For the interpolants to be valid: • The clause partitions Aand Bmust remain the same. • The global variables must remain the same.

46. Which nodes can be shared? f (x1,x2,…,xm): ( f )(CNFLeft)( f *)(CNFRight)(x1 = x1*)(x2 = x2*)…(xm = xm*) B A g (x1,x2,…,xm): ( g)(CNFLeft)( g *)(CNFRight)(x1 = x1*)(x2 = x2*)…(xm = xm*) B A

47. Which nodes can be shared? f (x1,x2,…,xm): ( f )(CNFLeft)( f *)(CNFRight)(x1 = x1*)(x2 = x2*)…(xm = xm*) B A Only the assertion clausesdiffer g (x1,x2,…,xm): ( g)(CNFLeft)( g *)(CNFRight)(x1 = x1*)(x2 = x2*)…(xm = xm*) B A

48. Restructuring Proofs • Color the assertion clauses and descendants black. • Color the remaining clauses white. • Resolve black nodes from white nodes.

49. Restructuring Proofs

50. Restructuring Proofs